Energy Dissipation and Regularity for a Coupled Navier-Stokes and Q-Tensor System

We study a complex non-newtonian fluid that models the flow of nematic liquid crystals. The fluid is described by a system that couples a forced Navier-Stokes system with a parabolic-type system. We prove the existence of global weak solutions in dimensions two and three. We show the existence of a Lyapunov functional for the smooth solutions of the coupled system and use the cancellations that allow its existence to prove higher global regularity, in dimension two. We also show the weak-strong uniqueness in dimension two

Colloquium: Mechanical formalisms for tissue dynamics

The understanding of morphogenesis in living organisms has been renewed by tremendous progressin experimental techniques that provide access to cell-scale, quantitative information both on theshapes of cells within tissues and on the genes being expressed. This information suggests that ourunderstanding of the respective contributions of gene expression and mechanics, and of their crucialentanglement, will soon leap forward. Biomechanics increasingly benefits from models, which assistthe design and interpretation of experiments, point out the main ingredients and assumptions, andultimately lead to predictions. The newly accessible local information thus calls for a reflectionon how to select suitable classes of mechanical models. We review both mechanical ingredientssuggested by the current knowledge of tissue behaviour, and modelling methods that can helpgenerate a rheological diagram or a constitutive equation. We distinguish cell scale ("intra-cell")and tissue scale ("inter-cell") contributions. We recall the mathematical framework developpedfor continuum materials and explain how to transform a constitutive equation into a set of partialdifferential equations amenable to numerical resolution. We show that when plastic behaviour isrelevant, the dissipation function formalism appears appropriate to generate constitutive equations;its variational nature facilitates numerical implementation, and we discuss adaptations needed in thecase of large deformations. The present article gathers theoretical methods that can readily enhancethe significance of the data to be extracted from recent or future high throughput biomechanicalexperiments.Comment: 33 pages, 20 figures. This version (26 Sept. 2015) contains a few corrections to the published version, all in Appendix D.2 devoted to large deformation

Dissipative Ordered Fluids: Theories for liquid crystals

This is a book on the dissipative dynamics of ordered fluids, with a particular focus on liquid crystals. It covers a whole range of different theories, mainly concerned with nematic liquid crystals in both their chiral and nonchiral variants. The authors begin by giving a detailed account of the molecular origins of orientational order in fluids

Flow and reorientation in the dynamics of nematic defects

We propose a simple phenomenological model for hydrodynamic defect dynamics in nematic liquid crystals, inspired by the Ericksen-Leslie theory. We identify the main forces that govern both fluid and defect motion and we comment on their symmetry. As shown for two annihilating disclinations, our model is predictive for arbitrary length scales and topological charges

Dynamics of dissipative ordered fluids

A variational principle is proposed that allows to derive the equations of motion for a fluid with a general microstructure described by a tensorial order parameter. The only constitutive ingredients are the densities of the free energy and the dissipation, both subject to appropriate invariance requirements. As an illustration, it is shown how the hydrodynamic theory for uniaxial nematic liquid crystals can be derived within this setting

Reorientational dynamics of conjugated nematic point defects

To appreciate the universal qualitative features of defect annihilation in nematic liquid crystals, we study how the viscous force of reorientational dynamics behaves under a transformation that reverses the sign of the defect's topological charge. As an illustration of our general results, we consider a class of point defects that were first studied by A. Saupe. The reorientational viscous forces acting on them differ dramatically from those acting on line defects

Steric effects in dispersion forces interactions

Classically, there have been two different ways to obtain mean-field theories for liquid crystals. One is based on short-range repulsive steric forces and the other on long-range attractive dispersion forces. In the former approach, it is the anisotropic shape of the molecules that leads to the anisotropic interaction, and in the latter it is the anisotropy of the molecular polarizability. In real molecules both causes of anisotropy can be expected to contribute to the effective interaction, and so it is desirable to assess the combined effect of anisotropic long-range attraction and short-range repulsion. Here we present an avenue to this end. We start from dispersion forces interactions and combine them with hard-core repulsions in a formal theory, whose crucial element is the steric tensor, a fourth-rank tensor depending on the anisotropy of the interacting molecules. This tensor can be determined analytically for a special class of molecular shapes

Fluids with dissipative microstructure

A variational principle is proposed which allows to derive the equations of motion for a dissipative fluid with general microstructure. The only constitutive ingredients are the densities of the free energy and the dissipation, both subject to appropriate invariance requirements. The strict interplay between the microstructures considered here and those studied by Capriz is also examined in some detail

Dynamics of kinks in biological membranes

We propose a two-dimensional model for the dynamics of the kinks created in a biological membrane by the interaction with a movable bead. We arrive at the evolution equations for both the bead and the membrane, whence we conclude that the force exerted on the bead by a fixed membrane points in the direction along which the curvature of the membrane is more concentrated. This is the first step towards understanding the basic mechanism behind the dynamics of protein aggregation which takes place on biological membranes